Journal of Computational and Applied Mathematics 233 (2010) 2438–2448

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Tempered stable Lévy motion and transient super-diffusionBoris Baeumer a,∗, Mark M. Meerschaert ba Department of Mathematics & Statistics, University of Otago, Dunedin, New Zealandb Department of Statistics & Probability, Michigan State University, Wells Hall, E. Lansing, MI 48824, United States

a r t i c l e i n f o

Article history:Received 23 September 2008Received in revised form 8 October 2009

Keywords:Fractional derivativesParticle trackingPower lawTruncated power law

a b s t r a c t

The space-fractional diffusion equation models anomalous super-diffusion. Its solutionsare transition densities of a stable Lévy motion, representing the accumulation of power-law jumps. The tempered stable Lévy motion uses exponential tempering to cool thesejumps. A tempered fractional diffusion equation governs the transition densities, whichprogress from super-diffusive early-time to diffusive late-time behavior. This articleprovides finite difference and particle trackingmethods for solving the tempered fractionaldiffusion equationwith drift. A temporal and spatial second-order Crank–Nicolsonmethodis developed, based on a finite difference formula for tempered fractional derivatives. Anew exponential rejection method for simulating tempered Lévy stables is presented tofacilitate particle tracking codes.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

The diffusion equation ∂tp = c∂2x p governs the transition densities of a Brownianmotion B(t), and its solutions spread atthe rate t1/2 for all time. The space-fractional diffusion equation ∂tp = c∂αx p for 0 < α < 2 governs the transition densitiesof a totally skewed α-stable Lévy motion S(t), and its solutions spread at the super-diffusive rate S(ct) ∼ c1/αS(t) for anytime scale c. The super-diffusive spreading is the result of large power-law jumps, with the probability of a jump longerthan r falling off like r−α [1–3]. The super-diffusive scaling of a stable Lévy motion cannot be expressed in terms of thesecond moment, which fails to exist due to the power-law tail of the probability density [4]. Instead one can use fractionalmoments [5] or quantiles.The totally skewed α-stable Lévy motion S(t) is the scaling limit of a random walk with power-law jumps: c−1/α(X1 +

· · · + X[ct]) ⇒ S(t) in distribution as the time scale c → ∞, when the independent jumps all satisfy P(X > r) = Ar−αfor large x > 0. Here 0 < α < 2 so that the second moment is infinite, and the usual Gaussian central limit theoremdoes not apply. The random walk with power-law jumps is called a Lévy flight [6]. The order of the fractional derivativein the governing equation ∂tp = c∂αx p equals the power-law index of the jumps. A more general jump variable withP(X < −r) = qAr−α and P(X > r) = (1 − q)Ar−α leads to a governing equation ∂tp = cq∂α−xp + (1 − q)c∂

αx p with

0 ≤ q ≤ 1. A continuous time randomwalk (CTRW) imposes a randomwaiting time Tn before the jumps Xn, and power-lawwaiting times P(T > t) = Bt−β for 0 < β < 1 lead to a space–time fractional governing equation ∂βt p = c∂αx p. See [7] formore details. Fractional diffusion equations are important in applications to physics [2,3], finance [8], and hydrology [9].Truncated Lévy flights were proposed by Mantegna and Stanley [10,11] as a modification of the α-stable Lévy motion,

since many deem the possibility of arbitrarily long jumps, and the mathematical fact of infinite moments, physicallyunrealistic. In that model, the largest jumps are simply discarded. Some other modifications to achieve finite secondmoments were proposed by Sokolov et al. [12], who add a higher-order power-law factor, and Chechkin et al. [13], who

∗ Corresponding author.E-mail addresses: bbaeumer@maths.otago.ac.nz (B. Baeumer), mcubed@stt.msu.edu (M.M. Meerschaert).

0377-0427/$ – see front matter© 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.cam.2009.10.027

B. Baeumer, M.M. Meerschaert / Journal of Computational and Applied Mathematics 233 (2010) 2438–2448 2439

add a nonlinear friction term. Tempered Lévy motion takes a different approach, exponentially tempering the probabilityof large jumps, so that very large jumps are exceedingly unlikely, and all moments exist [14]. Exponential tempering offerstechnical advantages, since the tempered process remains an infinitely divisible Lévy process whose governing equationcan be identified, and whose transition densities can be computed at any scale. Those transition densities solve a temperedfractional diffusion equation, quite similar to the fractional diffusion equation [15]. Like the truncated Lévy flights, theyevolve from super-diffusive early-time behavior, to diffusive late-time behavior.This paper provides finite difference and particle trackingmethods for solving the tempered fractional diffusion equation.

Higher-order methods for fractional diffusion equations are based on the Grünwald finite difference approximation for thefractional derivative [16,17]. This paper develops a second-order Crank–Nicolson scheme, using a variant of the Grünwaldfinite difference formula for tempered fractional derivatives combined with a Richardson extrapolation.Particle tracking solutions for fractional diffusion equations take a Langevin approach, simulating a Markov process

whose generator is the adjoint of the fractional derivative operator [18]. For tempered anomalous diffusion, particle trackingcan be accomplished using an exponential rejection method for simulating tempered stables. Since the tempered stableprocess is still an infinitely divisible Lévy process, increments can be simulated for any choice of time step size.

2. Tempered stable Lévy motion

The totally skewed α-stable Lévy motion x = S(t) for 0 < α < 2, α 6= 1 has a probability density p(x, t) that solves afractional diffusion equation:

∂tp(x, t) = c∂αx p(x, t). (1)

Take Fourier transforms (denoted by f (k) =∫e−ikxf (x) dx)) to get ddt p(k, t) = c(ik)

α p(k, t), recognizing that the fractionalderivative ∂αx corresponds to multiplication by (ik)

α in Fourier space. The point-source solution p(k, t) = etc(ik)αinverts

to a totally skewed α-stable density with no closed form, except in a few exceptional cases [2,19]. The constant c > 0 for1 < α < 2, and c < 0 for 0 < α < 1. The useful scaling property t1/αp(t1/αx, t) = p(x, 1) is evident from the Fouriertransform. The process has heavy tails with P(S(t) > r) ≈ Ar−α , so that any moment 〈S(t)n〉 =

∫xnP(x, t) dx of order

n > α diverges [4]; hence the need for tempering in some circumstances.The basic idea of exponential tapering is to modify the density function to the form e−λxp(x, t) and re-normalize: Lemma

2.2.1 on p. 67 in Zolotarev [20] shows that the Fourier transform∫e−iuxp(x, t) dx = etc(iu)

αhas an analytic extension

to =(u) < 0. Setting u = k − iλ shows that∫e−ikxe−λxp(x, t) dx = etc(λ+ik)

α. For the special case 0 < α < 1,

a simpler argument with Laplace transforms appears in Rosiński [14]. Since k = 0 yields the total mass, the densitye−tcλ

αe−λxp(x, t) integrates to one. The Fourier transform is easily computed: etc[(λ+ik)

α−λα ]. Exponential tapering changes

the mean: i ddketc[(λ+ik)α−λα ]

|k=0 = −ctαλα−1. Centering yields

pλ(k, t) = etc[(λ+ik)α−λα−ikαλα−1].

Invert the Fourier transform to get the mean-zero tempered stable density:

pλ(x, t) = e−λxect(α−1)λαp(x− ctαλα−1, t). (2)

Since ddt pλ(k, t) = c[(λ+ ik)α−λα− ikαλα−1]pλ(k, t), Fourier inversion reveals the tempered fractional diffusion equation

∂tpλ(x, t) = c ∂α,λx pλ(x, t) (3)

where ∂α,λx f (x) is the inverse Fourier transform of [(λ+ ik)α− λα − ikαλα−1]f (k). An easy calculation [15] shows that

∂α,λx f (x) = e−λx ∂αx [e

λx f (x)] − λα f (x)− αλα−1∂xf (x), (4)

which modifies the notation in [15] so that the tempered fractional diffusion equation (3) preserves the center of mass.Everything above carries through without modification for the case α = 2. In this case we can explicitly compute

p(x, t) = (4π tc)−1/2e−x2/(4tc). After substituting into (2), a little algebra shows that pλ(x, t) = p(x, t). Hence tempering

has no effect in the Gaussian case, and a tempered diffusion is exactly the same as a classical diffusion.The case α = 1 requires a different form p(k, t) = etc(ik) ln(ik) to solve (1) with c > 0. Now the scaling property

is tp(tx + tc ln t, t) = p(x, 1). Apply the exponential tempering, normalize and center as before to derive the Fourierrepresentation

pλ(k, t) = etc[(λ+ik) ln(λ+ik)−λ ln λ−ik(1+ln λ)]. (5)

Note that Lemma 2.2.1 in Zolotarev [20] also applies in this case. Invert to get the mean-zero tempered stable density

pλ(x, t) = e−λxetcλp(x− tc(1+ ln λ+ ln(tc)), t) (6)

that solves the tempered fractional diffusion equation (3), where ∂α,λx f (x) is the inverse Fourier transform of [(λ+ ik) ln(λ+ik)− λ ln λ− ik(1+ ln λ)]f (k)when α = 1.

2440 B. Baeumer, M.M. Meerschaert / Journal of Computational and Applied Mathematics 233 (2010) 2438–2448

To summarize, a totally positively skewed stable density with index 0 < α < 2 can be tempered so that moments of allorders exist. The positively skewed tempered stable density is obtained by: (i) multiplying the stable density by e−λx; (ii)rescaling by a constant that makes the resulting function integrate to one; and (iii) shifting by a constant that makes theresulting density function have mean zero. Then we have the following result.

Proposition 1. Let S(t) be a totally skewed α-stable Lévy process for 0 < α < 2 with density p(x, t) and Fourier transform

E[e−ikS(t)] = p(k, t) ={exp [ct(ik)α] α 6= 1exp [ctik ln(ik)] α = 1. (7)

Then the tempered α-stable Lévy process Sλ(t) has density

pλ(x, t) ={e−λx+(α−1)ctλ

αp(x− ctαλα−1, t) α 6= 1

e−λx+ctλp(x− ct(1+ ln λ), t) α = 1.(8)

Next we extend Proposition 1 to the general case, to accommodate both positive and negative particle jumps. If S(t) ispositively skewed, then −S(t) is negatively skewed, with density p(−x, t). For 0 < α < 2, α 6= 1 the density solves afractional diffusion equation ∂tp = c∂α−xp where ∂

α−x corresponds to multiplication by (−ik)

α in Fourier space. Temperingmultiplies the density by eλx, normalizes to total mass one, and centers, as before. The resulting tempered stable densityhas Fourier transform etc[(λ−ik)

α−λα+ikαλα−1] which just replaces ik by −ik in the formula (5). It solves the tempered space-

fractional diffusion equation ∂tp = c ∂α,λ−x pwhere ∂

α,λ−x f (x) is the inverse Fourier transform of [(λ− ik)α−λα+ ikαλα−1]f (k).

A similar sign change yields the negatively skewed tempered stable density and governing equation in the case α = 1. Forthe general case, generate two independent positively skewed tempered stable processes S+λ (t) and S

−

λ (t) with the sameindex α, and set

Sλ(t) = qS−λ (t)+ (1− q)S+

λ (t) (9)

where 0 ≤ q ≤ 1. The probability densities of this process solve the general tempered space-fractional diffusion equation

∂tp = cq ∂α,λ−x p+ c(1− q) ∂

α,λx p. (10)

Another interesting characterization of the tempered stable process can be stated in terms of its Lévy representation. Inshort, a Lévy process is the limiting case of a compound Poisson process whose jumps are described by the Lévy measure(e.g., see [21, Remark 3.1.18], and (38)). The remainder of this section is devoted to showing that the tempered stable processcan also be defined in terms of an exponentially tempered Lévy measure. In the process, we will also show that exponentialtempering of the density function can also be used for any Lévy process with positive jumps.Any infinitely divisible process S(t) on R can be characterized by its Fourier transform

E[e−ikS(t)] = exp[t(−aik− σ 2k2 +

∫x6=0

(e−ikx − 1+

ikx1+ x2

)φ(dx)

)](11)

where a ∈ R, σ ≥ 0 and the Lévy measure φ with∫ x2

1+x2φ(dx) <∞, are uniquely determined [4,21]. We call [a, σ , φ] the

Lévy representation.Given an infinitely divisible process S(t)with Lévy representation [0, 0, φ], we define the exponentially tempered Lévy

measure φλ(dx) = exp(−λ|x|)φ(dx). It is clear that∫ x2

1+x2φλ(dx) <∞, so that there exists another infinitely divisible Lévy

process Sλ(t) with Lévy representation [a, 0, φλ] for any a ∈ R. Write φ = φ+ + φ− where φ+(U) = φ(U ∩ {x : x > 0})is the positive part of the Lévy measure. It is clear from (11) that we can write S(t) = S+(t) + S−(t) where S+(t) isinfinitely divisible with Lévy representation [0, 0, φ+], S−(t) is infinitely divisible with Lévy representation [0, 0, φ−], andS+(t), S−(t) are independent. In order to develop a theory of tempered infinitely divisible laws, it suffices to consider justthe positive case. The extension to the signed case is the same as for tempered stable laws. Hence it suffices to considerinfinitely divisible laws whose Lévy measure is concentrated on the positive real line.

Theorem 2. Let S(t) be an infinitely divisible process with Lévy representation [a, σ , φ], where φ is supported on R+. Write theFourier transform E[e−ikS(t)] = exp(tΦ(k)) and the probability distribution P(U, t) =

∫U P(dx, t) = P(S(t) ∈ U) for Borel

measurable U ⊂ R. Then Φ has an analytic extension to the lower half plane. Given λ > 0, define an infinitely divisible processSλ(t)with Lévy representation [d, σ , φλ]where φλ(dx) = e−λxφ(dx). The mean E[Sλ(t)] exists for all λ > 0, and we can choosed ∈ R so that E[Sλ(t)] = 0. The resulting tempered infinitely divisible Lévy process Sλ(t) has probability distribution

Pλ(dx, t) = e−λx−Φ(−iλ)t−iλΦ′(−iλ)tP(dx+ iΦ ′(−iλ)t, t). (12)

Proof. First we show that

Φ(k) = −aik− σ 2k2 +∫∞

0

(e−ikx − 1+

ikx1+ x2

)φ(dx)

B. Baeumer, M.M. Meerschaert / Journal of Computational and Applied Mathematics 233 (2010) 2438–2448 2441

has an analytic extension to the lower half plane. For =(z) < 0, let

Ψ (z) =∫∞

0

1+ x2

x2

(e−izx − 1+

izx1+ x2

)x2

1+ x2φ(dx).

As φ is a Lévy measure,∫∞

0x2

1+x2φ(dx) <∞. The rest of the integrand satisfies∣∣∣∣1+ x2x2

(e−izx − 1+

izx1+ x2

)∣∣∣∣ = ∣∣∣∣e−izx − 1+ izxx2+ e−izx − 1

∣∣∣∣<

{|z2| exp(|z|x)+ 2 x < 1(2+ |z|x)/x2 + 2 x ≥ 1.

(13)

Hence Ψ is well defined, its complex derivative

limw→0

1w

∫∞

0

1+ x2

x2

(e−i(z+w)x − 1+

i(z + w)x1+ x2

−

(e−izx − 1+

izx1+ x2

))x2

1+ x2φ(dx)

= limw→0

1w

∫∞

0

(e−izx

(e−iwx − 1

)+ iwx

x2+ e−izx

(e−iwx − 1

)) x2

1+ x2φ(dx)

= −i∫∞

0

(e−izx − 1x

+ xe−izx)

x2

1+ x2φ(dx) (14)

exists by the Dominated Convergence Theorem as∣∣∣∣∣ 1w e−izx(e−iwx − 1

)+ iwx

x2+ e−izx

e−iwx − 1w

∣∣∣∣∣ <∣∣∣∣∣ 1w e−izx

(e−iwx − 1

)+ iwx

x2

∣∣∣∣∣+∣∣∣∣e−izx e−iwx − 1w

∣∣∣∣ . (15)

The second term is bounded by |e−izx|xe|w|x < xe(=(z)+|w|)x < M for =(z) + |w| < 0. For x > 1, almost the same boundapplies to the first term. For x < 1, a Taylor expansion shows that∣∣∣∣∣e−izx

(e−iwx − 1

)+ iwx

wx2

∣∣∣∣∣ < M. (16)

Note that

Ψ (k− iλ)− Φ(k)− aik− σ 2k2 =∫∞

0e−ikx−λx − e−ikx −

λx1+ x2

φ(dx)

=

∫∞

0e−ikx

(e−λx − 1

)−

λx1+ x2

φ(dx) (17)

tends to zero as λ→ 0+ by another dominated convergence argument, similar to (15) without the 1/w. HenceΨ (z)−aiz−σ 2z2 is the analytic extension ofΦ . In other words,

Φ(k− iλ) = −ai(k− iλ)− σ 2(k− iλ)2 +∫∞

0

(e−(ik+λ)x − 1+

(ik+ λ)x1+ x2

)φ(dx)

is the unique extension ofΦ .Now∫

∞

0

(e−ikx − 1+

ikx1+ x2

)e−λx φ(dx) = Φ(k− iλ)− Φ(−iλ)+ aik+ σ 2k2 − 2ikλσ 2 + bik (18)

for b =∫∞

0 (e−λx− 1) x

1+x2φ(dx) ∈ R. Let H(dx, t) denote the probability distribution of the infinitely divisible process with

Lévy representation [a, σ , φλ], and denote its Fourier transform by H(k, t) =∫e−ikxH(dx, t). Then

H(k, t) = exp[t(Φ(k− iλ)− Φ(−iλ)− 2ikλσ 2 + bik

)].

As H is infinitely differentiable, by [22, Prop. 5.1.19] all moments are finite. The mean is given by tµ = i ∂∂k H(0, t) =

itΦ ′(−iλ) + 2tλσ 2 − bt . Setting d = a − µ yields an infinitely divisible process Sλ(t) with Lévy representation [d, σ , φλ]satisfying E[Sλ(t)] = 0 for all t > 0. Its probability distribution Pλ(dx, t) has Fourier transform

Pλ(k, t) = exp[t(Φ(k− iλ)− Φ(−iλ)+ ikiΦ ′(−iλ)

)]. (19)

2442 B. Baeumer, M.M. Meerschaert / Journal of Computational and Applied Mathematics 233 (2010) 2438–2448

By [23, Thm. III.10], the moment sequence {µn}∞0 with µn = in ∂n∂kn pλ(0, t) is positive (in the sense of [23, Def. III.9b]). Let

q(s) = Pλ(−is, t). Then µn = (−1)n dn

dsn q(0) and hence by [23, Thm. VI.19.c], the integral

q(s) = Pλ(−is, t) =∫∞

−∞

e−sx Pλ(dx, t)

exists for some measure Pλ and all<(s) > −λ. Therefore

Pλ(k− iν, t) =∫∞

−∞

e−ikxe−νx Pλ(dx, t)

for all k ∈ R and ν > −λ. As by (19)

limν→−λ+

Pλ(k− iν, t) = exp[t(Φ(k)− Φ(−iλ)+ (ik− λ)iΦ ′(−iλ)

)]we obtain, using the shift formula for the Fourier transform, that the inverse Fourier transforms are the same as well:

eλxPλ(dx, t) = e−Φ(−iλ)t−iλΦ′(−iλ)tP(dx+ iΦ ′(−iλ)t, t),

which concludes the proof. �

Remark. If S(t) has a Lebesgue density p(x, t), so that P(dx, t) = p(x, t)dx, then it follows immediately from Theorem 2that the tempered process Sλ(t) has density

pλ(x, t) = e−λx−Φ(−iλ)t−iλΦ′(−iλ)tp(x+ iΦ ′(−iλ)t, t). (20)

If we takeΦ(k) = c(ik)α for α 6= 1, orΦ(k) = c(ik) log(ik) for α = 1, then it follows by an easy computation that (8) holds.This shows that exponentially tempering the stable density is equivalent to exponentially tempering the correspondingLévy measure. More generally, (20) shows that exponentially tempering the Lévy measure of an infinitely divisible law isequivalent to exponentially tempering the density.

3. Finite difference methods

In this section we develop a second-order finite difference method to solve the tempered fractional advection dispersionequation with drift on a bounded interval [xL, xR]with Dirichlet boundary conditions:

∂tu(x, t) = −v(x)∂xu(x, t)+ c(x)∂α,λx u(x, t)+ q(x, t) (21)

with u(x, 0) = u0(x), u(xR, t) = BR(t), and u(x, t) = 0 for all x < xL and t > 0. We assume v(x) ≥ 0, c(x) ≥ 0 and1 < α ≤ 2, to be consistent with the fractional advection dispersion equation for left-to-right flow [16]. Our approach issimilar to [17]. First we develop an implicit Crank–Nicolson scheme that is second order in time, but fails to be second orderin space, due to the fact that the finite difference approximation of the fractional derivative is only first order. Thenwe applyRichardson extrapolation to obtain amethod that is second order in both variables.We prove second-order consistency, andstability, and conclude with a numerical example.Recall the fractional binomial formula

∞∑k=0

(αk

)zk = (1+ z)α (22)

which is valid for any α > 0 and complex |z| ≤ 1. Define

wk =Γ (k− α)

Γ (−α)Γ (k+ 1)= (−1)k

(αk

)and note thatwk can be computed recursively viaw0 = 1, w1 = −α,wk+1 = k−α

k+1wk. Our first result establishes a suitablefinite difference approximation for the tempered fractional derivative. Recall that Lp denotes the class of functions f forwhich

∫|f (x)|p dx exists, and the Sobolev spaceW k,p contains the Lp functions whose derivatives of order j = 1, 2, . . . , k

are also in Lp.

Proposition 3. Let f ∈ W n+3,1 for some integer n ≥ 1. Let p ∈ R, h > 0, λ ≥ 0 and 1 < α ≤ 2. Define

1αh,pf (x) :=1hα

∞∑j=0

(wje−(j−p)hλf (x− (j− p)h)

)− ephλ

(1− e−hλ

)αhα

f (x). (23)

B. Baeumer, M.M. Meerschaert / Journal of Computational and Applied Mathematics 233 (2010) 2438–2448 2443

Then there exist constants aj independent of h, f , x and λ such that

1αh,pf (x) = ∂α,λx f (x)+ αλ

α−1f ′(x)+n−1∑j=1

aj(e−λx∂α+jx

[eλxf (x)

]− λα+jf (x)

)hj + O(hn) (24)

uniformly in x ∈ R.

Proof. The proof is similar to [17, Proposition 3.1]. Use the fractional binomial formula (22) with z = −e−h(λ+ik), along withthe fact that e−ika f (k) is the Fourier transform of f (x− a), to see that (23) has Fourier transform φh(k)f (k)with

φh(k) =1hα

(∞∑j=0

wje−(j−p)h(λ+ik) − ephλ(1− e−hλ

)α)

= eph(λ+ik)(1− e−h(λ+ik)

h

)α− ephλ

(1− e−hλ

h

)α= (λ+ ik)αωh(λ+ ik)− λαωh(λ) (25)

where

ωh(z) = ephz(1− e−hz

hz

)α.

A Taylor series expansion and some elementary estimates imply∣∣∣∣∣ωh(λ+ ik)− n−1∑j=0

aj(λ+ ik)jhj∣∣∣∣∣ ≤ Chn|λ+ ik|nephλ

for some constants aj and C independent of h. Hence

φh(k)f (k) =n−1∑j=0

aj((λ+ ik)α+j − λα+j

)hj f (k)+ ψ(k, h)

for some function ψ with

|ψ(k, h)| ≤ Chn(|λ+ ik|α+n + λα+n)ephλ|f (k)|.

Since f ∈ W n+3,1, a standard argument using the Riemann–Lebesgue Lemma implies that |f (k)| ≤ C(1 + |k|)−n−3. Hencek 7→ ψ(k, h) ∈ L1(R) and |ψ(x, h)| ≤ C |h|nephλ for all x ∈ R, and then Fourier inversion yields (24) uniformly overx ∈ R. �

Remark 4. Note that the approximation (24) has terms involving ∂α+jx f (x). For these to be uniformly bounded, f has to besufficiently regular, which is the case if f ∈ W n+3,1. However, if for example f (x) = xβ , then ∂α+jx f (x) is not bounded nearx = 0 if α + j > β , and (24) will not hold. One should be able to rectify this by using the starting quadrature weights ofLubich [24], but we have not yet attempted this.

Next we outline a Crank–Nicolson scheme for solving the tempered fractional diffusion equation (21) on x ∈ [xL, xR]and t ∈ [0, T ]. Define tn = n1t to be the integration time 0 ≤ tn ≤ T and 1x = h > 0 to be the spatial grid size withxi = xL + i1x, i = 0, . . . ,Nx. Define uni = u(xi, tn), ci = c(xi), vi = v(xi), and q

n+1/2i = q(xi, (tn+1 + tn)/2). Let Uni define

the numerical approximation to the exact solution uni . In view of Proposition 3 we will write

Un+1i − Uni1t

=

(−vi + ciαλα−1

2δx +

ci2δα,λx

) (Un+1i + Uni

)+ qn+1/2i , (26)

where δxUni = (Uni − U

ni−1)/1x and

δα,λx Uni =

1(1x)α

(i+1∑k=0

wke−(k−1)λ1xUni−k+1

)−eλ1x

(1x)α(1− e−λ1x

)αUni . (27)

Rearranging (26) for Un+1i in matrix form yields

(I −1tA)Un+1 = (I +1tA)Un + Q n+1/21t, (28)

2444 B. Baeumer, M.M. Meerschaert / Journal of Computational and Applied Mathematics 233 (2010) 2438–2448

where

Un = [Un1 , . . . ,UnNx−1]

T ,

Q n+1/2 = [qn+1/21 , qn+1/22 , . . . , qn+1/2Nx−1 + (ηNx−1eλ1x)(Bn+1R + B

nR)]T , (29)

ηi =ci

2(1x)α and I is the (Nx − 1)× (Nx − 1) identity matrix. The entries of the matrix A are collected from (26) to yield

Ai,j =

ηie−λ(i−j)1xwi−j+1 j ≤ i− 2

ηie−λ1xw2 +ciαλα−1 + vi21x

j = i− 1

ηi(w1 − eλ1x

(1− e−λ1x

)α)−ciαλα−1 + vi21x

j = i

ηieλ1x j = i+ 10 j > i+ 1

(30)

for i, j = 1, . . . ,Nx − 1. The matrix A is lower triangular plus entries on the first super-diagonal j = i + 1. Note that theboundary condition u(xNx , tn) = BR(tn) = B

nR is incorporated in the forcing term Q , essentially to account for the fact that

the super-diagonal term is cut off in the bottom row.

Theorem 5. The Crank–Nicolson scheme (28) is consistent.

Proof. The proof follows immediately from Proposition 3, along with the well-known consistency for the first-orderdifference approximation δxUni = (U

ni − U

ni−1)/1x. �

Proposition 6. The fractional Crank–Nicolson scheme (28) is unconditionally stable.

Proof. The proof is similar to [17, Proposition 3.3]. Note that (22) implies∞∑j=0

wje−(j−1)λ1x = eλ1x(1− e−λ1x)α.

It follows by a straightforward calculation that∑i+1j=−∞ Ai,j = 0. Since the only negative entries of A are on the diagonal,

this implies that −Ai,i >∑i+1j=0,j6=i Ai,j. Hence the matrix is diagonally dominant and using the Gershgorin theorem

[25, pp. 135–136], all the eigenvalues of the matrix A in (30) have negative real parts. Then the spectral mapping theoremshows that the eigenvalues of (I−1tA)−1(I+1tA) have complex absolute value less than 1, so the system is unconditionallystable. �

Some standard arguments along with Proposition 3 show that the Crank–Nicolson scheme is O(1t2+1x). We can improvethe order of approximation by using Richardson extrapolation to obtain second-order convergence; i.e., solve the systemtwice, once with grid size1x and again with grid size1x/2. Let Ui = 2U2i,1x/2 − Ui,1x be the extrapolated solution on thecoarse grid. Then, in view of Proposition 3, the extrapolated scheme is O(1t2 +1x2).

Example 7. Consider the tempered fractional diffusion equation with no drift

∂tu(x, t) = c(x)(e−λx

∂α

∂xαeλxu(x, t)− λαu(x, t)− αλα−1

∂

∂xu(x, t)

)+ q(x, t)

with x ∈ [0, 1], α = 1.6, λ = 2, initial condition

u(x, 0) =xβe−λx

Γ (β + 1),

with β = 2.8, diffusion coefficient

c(x) =xαΓ (1+ β − α)

Γ (β + 1),

forcing function

q(x, t) = e−λx−tΓ (1+ β − α)Γ (β + 1)

((1− α)λαxα+β

Γ (β + 1)+αβλα−1xα+β−1

Γ (β)−

2xβ

Γ (1+ β − α)

)and boundary condition

B(t) = u(1, t) =e−λ−t

Γ (β + 1).

B. Baeumer, M.M. Meerschaert / Journal of Computational and Applied Mathematics 233 (2010) 2438–2448 2445

Table 1Maximum error behavior for Crank–Nicolson (CN) and extrapolated (CNX) schemes showing improved second-order convergence.

1t 1x CN max error CN error rate CNX max error CNX error rate

1/10 1/50 7.7738× 10−5 – 2.8514× 10−6 –1/20 1/100 3.8353× 10−5 2.03 7.2120× 10−7 3.951/40 1/200 1.9055× 10−5 2.01 1.8157× 10−7 3.971/80 1/400 9.4976× 10−6 2.01 4.5555× 10−8 3.99

The exact solution is given by

u(x, t) =xβe−λx−t

Γ (1+ β).

Wesolved the equation numerically to time t = 1with andwithout extrapolation. The extrapolationwas done by halvingthe grid size to1x/2 while keeping1t to obtain U2i,1x/2 and then computing Ui = 2U2i,1x/2 − Ui,1x. The maximum error isgiven in Table 1.

4. Monte Carlo simulation

Since the tempered α-stable Lévy process Sλ(t) is infinitely divisible, it can be approximated by a random walk at anygivenmesh1t . In this section, we provide a general method to generate tempered stable random variables. More generally,the method can be used to simulate any exponentially tempered random variable.Given a non-negative random variable X with density f (x), define the exponentially tempered density fλ(x) =

e−λxf (x)/∫∞

0 e−λuf (u) du. Take Y exponentially distributedwithmeanλ−1, independent of X . Generate (Xi, Yi) independent

and identically distributed with (X, Y ), and let N = min{n : Xn ≤ Yn}. Then XN has the exponentially tempered densityfunction fλ(x): To see this, let q = P(X ≤ Y ) =

∫∞

0 P(X ≤ y)λe−λy dy and compute

P(XN ≤ x) =∞∑n=1

P(XN ≤ x,N = n)

=

∞∑n=1

P(Xn ≤ x, Xn ≤ Yn)P(Xn−1 > Yn−1) · · · P(X1 > Y1)

=

∞∑n=1

P(Xn ≤ x, Xn ≤ Yn)(1− q)n−1

= q−1P(Xn ≤ x, Xn ≤ Yn) (31)

where

P(Xn ≤ x, Xn ≤ Yn) =∫∞

0P(X ≤ x, X ≤ y)λe−λy dy

=

∫ x

0P(X ≤ y)λe−λy dy+ P(X ≤ x)e−λx (32)

and hence the density of XN is

ddx

P(XN ≤ x) = q−1ddx

[∫ x

0P(X ≤ y)λe−λy dy+ P(X ≤ x)e−λx

]= q−1

[P(X ≤ x)λe−λx + f (x)e−λx − P(X ≤ x)λe−λx

](33)

which reduces to fλ(x), since q =∫∞

0 f (y)e−λy dy via integration by parts.

If S(t) is stable with index α < 1 and its density has Fourier transform p(k, t) = etc(ik)αthen S(t) ≥ 0, and the method

of [26,19] can be used to simulate S(t) = Z: Take U uniform on [−π/2, π/2],W exponential with mean 1, and set

Z = (|c|t)1/αsinα(U + π

2 )

cos(U)1/α

(cos(U − α(U + π

2 ))

W

)(1−α)/α. (34)

The tempered stable random variable Sλ(t) can be simulated by the rejection method: Draw Z , and Y exponential withmean λ−1. If Y < Z , reject and draw again; otherwise, set Sλ(t) = Z + ctαλα−1. To simulate the entire path of the process,write t = n1t and note that Sλ(t) =

∑nk=1[Sλ(k1t) − Sλ((k − 1)1t)], a random walk with all n summands independent

and identically distributed with Sλ(1t).

2446 B. Baeumer, M.M. Meerschaert / Journal of Computational and Applied Mathematics 233 (2010) 2438–2448

When α > 1, the stable density p(x, t) with Fourier transform p(k, t) = etc(ik)αis positive for all−∞ < x < ∞ and all

t > 0; i.e., the support is all of R. The exponentially tempered density pλ(x, t) = e−tcλαe−λxp(x, t) is still integrable because

p(x, t) → 0 faster than any exponential function as x → −∞ [20, Theorem 2.5.2 and Eq. (2.5.18–19)]. To simulate theserandom variables, we adapt the procedure for α < 1 and use the following theorem. Recall that for two random variablesX, Z with cumulative distribution functions F(x) = P(X ≤ x),G(z) = P(Z ≤ z), the Kolmogorov–Smirnov distance isdefined by d(X, Z) = sup{|F(x)− G(x)| : x ∈ R}.

Theorem 8. Let X be a random variable with density f (x) such that the integral I =∫∞

−∞e−λxf (x) dx exists. Define the

exponentially tempered density fλ(x) = e−λxf (x)/I . Let Y be exponentially distributedwithmeanλ−1, independent of X. Generate(Xi, Yi) independent and identically distributed with (X, Y ), and for a ≥ 0 let Na = min{n : Xn ≤ Yn − a}. Then the density ofXNa converges in L

1 to fλ as a→∞, and hence d(XNa , X)→ 0 in the Kolmogorov–Smirnov distance.

Proof. Set q = P(X ≤ Y − a) and compute Fa(x) = P(XNa ≤ x) = q−1P(X ≤ x, X ≤ Y − a). Note that Fa(x) = q−1P(X ≤ x)

for x < −a, and, similar to (32), for x ≥ −awe have

Fa(x) = q−1(∫ x+a

0P(X ≤ y− a)λe−λy dy+ P(X ≤ x)e−λ(x+a)

).

Then the density of XNa is fa(x) = q−1f (x) for x < −a and, by the same argument as (33), fa(x) = q−1f (x)e−λ(x+a) for x ≥ −a.

Note that

q =∫∞

0P(X ≤ y− a)λe−λy dy = e−λa

∫∞

−aP(X ≤ y)λe−λy dy

=

∫−a

−∞

f (y) dy+ e−λa∫∞

−af (y)e−λy dy.

Let ha = eλa∫−a−∞f (y) dy and ga =

∫−a−∞e−λyf (y) dy. Then q = e−λa(I − ga + ha), 0 ≤ ha ≤ ga → 0 and the L1 distance∫

∞

−∞

|fλ(x)− fa(x)| dx =∫−a

−∞

∣∣∣∣e−λxf (x)I−f (x)q

∣∣∣∣ dx+ ∫ ∞−a

∣∣∣∣e−λxf (x)I−f (x)e−λ(x+a)

q

∣∣∣∣ dx=

∫−a

−∞

∣∣∣∣e−λxf (x)I−

eλaf (x)I − ga + ha

∣∣∣∣ dx+ ∫ ∞−a

∣∣∣∣e−λxf (x)I−f (x)e−λx

I − ga + ha

∣∣∣∣ dx≤

2gaI − ga + ha

+ (I − ga)(

1I − ga + ha

−1I

)≤

3gaI − ga + ha

→ 0.

Finally we note that

d(XNa , X) = supx∈R|Fλ(x)− Fa(x)|

= supx∈R

∣∣∣∣∫ x

−∞

fλ(y) dy−∫ x

−∞

fa(y) dy∣∣∣∣

≤ supx∈R

∫ x

−∞

|fλ(y)− fa(y)| dy

=

∫∞

−∞

|fλ(y)− fa(y)| dy ≤3ga

I − ga + ha→ 0 (35)

which completes the proof. �

Note that if P(X ≥ −a) = 1, then obviously ga = 0 and XNa has the exponentially tempered function fλ(x) as its density.To simulate Sλ(t) for α > 1, choose a > 0 so that P(S(t) < −a) is small, draw Z using

Z = (ct)1/αsinα(U − π

2 )

cos(U)1/α

(cos(U − α(U − π

2 +πα))

W

)(1−α)/α(36)

and Y exponential withmean λ−1. If Z > Y−a, reject and draw again; otherwise, set Sλ(t) = Z+ctαλα−1. Note that a largera increases accuracy of the simulationmethod, however a larger awill result in more rejections, increasing computing time.For α = 1, use the same method and set Sλ(t) = Z + ct(1+ ln λ)where

Z = (ct)(ln(π2ct)+

(π2+ U

)tan(U)− ln

( π2W cosUπ2 + U

)). (37)

B. Baeumer, M.M. Meerschaert / Journal of Computational and Applied Mathematics 233 (2010) 2438–2448 2447

0 5000 10000–150

–100

–50

0

λ = 0.1

0 5000 10000–200

0

200λ = 0.01

0 5000 10000–500

0

500λ = 0.001 λ = 0.0001

0 5000 10000

–500

0

500

1000

Fig. 1. Tempered stable Lévy motion Sλ(t)with α = 1.2.

A random walk simulation of the particle path can be accomplished as in the case α < 1, by simulating and addingindependent and identically distributed random jumps Sλ(1t). Fig. 1 illustrates a random walk simulation of Sλ(t) withα = 1.2, c = 1, 1t = 1, and 0 ≤ t ≤ T = 100,000 for four different values of the tempering parameter λ. Thesimulation produced no Z < −2.5, and the chosen shift a = 4 lead to few rejects [4000 out of 100,000 for λ = 0.1; 350for λ = 0.01; 30 for λ = 0.001; and 5 for λ = 0.0001]. In this case, the random variable S(1) is stable with index α, meanzero, skewness+1, and scale σ = (− cos(πα/2))1/α ≈ 0.3578 in the parameterization of Samorodnitsky and Taqqu [19].The Kolmogorov–Smirnov bound (35) is less than 10−14 (zero to machine precision) which can be verified by numericalintegration using the fast efficient codes of Nolan [27] for the stable density. Observe that as λ gets smaller, the anomalouslarge jumps in Fig. 1 increase, leading to super-diffusion. Particle tracking codes can be accomplished by simulating a largenumber of particles, and generating a histogram of the results at different time points. The particle tracking method can beextended to variable coefficient equations with complicated geometry and boundary conditions.An alternative simulation method uses the Poisson approximation for an infinitely divisible Lévy process. It follows from

results in Cartea and del Castillo-Negrete [15] that the density pλ(x, t) of Sλ(t) has Fourier transform

pλ(k, t) = exp[c0t∫∞

0(e−ikx − 1+ ikx)e−λxαx−α−1 dx

]where c0 = c(α−1)/Γ (2−α). Note that this corrects the constant in [15, Eq. (21)]. Given ε > 0 small, we estimate pλ(k, t)by

exp[c0t∫∞

ε

(e−ikx − 1+ ikx)e−λxαx−α−1 dx]= exp

[mεc0t(fε(k)− 1+ ikµε)

](38)

where mε =∫∞

εe−λxαx−α−1 dx, fε(x) = m−1ε e

−λxαx−α−1 is a probability density, and µε =∫∞

εxfε(x) dx is its mean.

Recognizing that the last line in (38) is the Fourier transform of a mean-centered compound Poisson [21], we can write

Sλ(t) ≈Nε(t)∑i=1

(Ji − µε)

where Ji are independent random variables with density fε , andNε(t) is a Poisson process with ratemεc0. The approximationbecomes exact as ε → 0. To simulate Ji, use the exponential rejection method for positive random variables X developedin this section, with P(X > x) = (x/ε)−α , so that X = ε/U1/α with U uniform on [0, 1]. To simulate the Poisson process,let Wi = ln(Ui)/(mεc0), Tn = W1 + · · · + Wn, and Nε(t) = max{n : Tn ≤ t}. Note that this approximation is actuallya continuous time random walk [2,3] with exponentially tempered power-law jumps. Setting λ = 0 yields a well-knownapproximation to the anomalous diffusion process S(t). In the special case α < 1, a more refined version of this simulationidea was developed in a recent paper of Cohen and Rosiński [28]. Their method reproduces the exact sample path of theprocess up to small jumps, using a series representation based on a similar Poissonian representation.

Acknowledgements

Wewould like to thankM. Kovaćs and the anonymous reviewers for the fruitful discussions and suggestions on improvingthe manuscript. M.M. Meerschaert was partially supported by NSF grants DMS-0803360 and EAR-0823965. This paper wascompletedwhile B. Baeumerwas on sabbatical leave at the Department of Statistics & Probability,Michigan State University.

2448 B. Baeumer, M.M. Meerschaert / Journal of Computational and Applied Mathematics 233 (2010) 2438–2448

References

[1] M.M. Meerschaert, D.A. Benson, B. Bäumer, Multidimensional advection and fractional dispersion, Phys. Rev. E 59 (1999) 5026–5028.[2] R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339 (2000) 1–77.[3] R. Metzler, J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractionaldynamics, J. Phys. A 37 (2004) R161–R208.

[4] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II, 2nd ed., Wiley, New York, 1971.[5] C. Nikias, M. Shao, Signal Processing with Alpha Stable Distributions and Applications, Wiley, New York, 1995.[6] M.F. Shlesinger, G. Zaslavsky, U. Frisch (Eds.), Lévy Flights and Related Topics in Physics, Springer, Berlin, 1995.[7] M.M. Meerschaert, H.P. Scheffler, Limit theorems for continuous time random walks with infinite mean waiting times, J. Appl. Probab. 41 (2004)623–638.

[8] E. Scalas, Five years of Continuous-Time RandomWalks in Econophysics. in: A. Namatame (Ed.), Proceedings of WEHIA 2004, Kyoto, 2004.[9] R. Schumer, M.M. Meerschaert, B. Baeumer, Fractional advection-dispersion equations for modeling transport at the Earth surface, J. Geophys. Res.(2009) in press (doi:10.1029/2008JF001246). Preprint available at: www.stt.msu.edu/~mcubed/fADEreview.pdf.

[10] R.N. Mantegna, H.E. Stanley, Stochastic process with ultraslow convergence to a Gaussian: The truncated Lévy flight, Phys. Rev. Lett. 73 (1994)2946–2949.

[11] R.N. Mantegna, H.E. Stanley, Scaling behavior in the dyamics of an economic index, Nature 376 (1995) 46–49.[12] I.M. Sokolov, A.V. Chechkin, J. Klafter, Fractional diffusion equation for a power-law-truncated Lévy process, Physica A 336 (2004) 245–251.[13] A.V. Chechkin, V.Yu. Gonchar, J. Klafter, R. Metzler, Natural cutoff in Lévy flights caused by dissipative nonlinearity, Phys. Rev. E 72 (2005) 010101.[14] J. Rosiński, Tempering stable processes, Stochastic Process. Appl. 117 (2007) 677–707.[15] Á. Cartea, D. del Castillo-Negrete, Fluid limit of the continuous-time random walk with general Lévy jump distribution functions, Phys. Rev. E 76

(2007) 041105.[16] M.M.Meerschaert, C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math. 172 (2004)

65–77.[17] C. Tadjeran, M.M. Meerschaert, H.-P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput.

Phys. 213 (2006) 205–213.[18] Y. Zhang, D.A. Benson, M.M. Meerschaert, H.-P. Scheffler, On using random walks to solve the space-fractional advection-dispersion equations, J.

Statist. Phys. 123 (2006) 89–110.[19] G. Samorodnitsky, M. Taqqu, Stable Non-Gaussian Random Processes, Chapman and Hall, New York, 1994.[20] V.M. Zolotarev, One-dimensional stable distributions, in: Translations of Mathematical Monographs, vol. 65, American Mathematical Society,

Providence, RI, 1986; Translated from the Russian by H. H. McFaden, Translation edited by Ben Silver.[21] M.M. Meerschaert, H.-P. Scheffler, Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice, John Wiley &

Sons Inc., New York, 2001.[22] P.L. Butzer, R.J. Nessel, Fourier analysis and approximation, in: Volume 1: One-dimensional Theory, in: Pure and Applied Mathematics, vol. 40,

Academic Press, New York, 1971.[23] D.V. Widder, The Laplace Transform, 2nd ed., in: Princeton Mathematical Series, vol. 6, Princeton University Press, Princeton, NJ, 1946.[24] Ch. Lubich, Discretized fractional calculus, SIAM J. Math. Anal. 17 (1986) 704–719.[25] E. Isaacson, H.B. Keller, Analysis of Numerical Methods, John Wiley & Sons Inc., New York, 1966.[26] J.M. Chambers, C.L. Mallows, B.W. Stuck, A method for simulating stable random variables, J. Amer. Statist. Assoc. 71 (1976) 340–344.[27] J.P. Nolan, An algorithm for evaluating stable densities in Zolotarev’s (M) parameterization, Math. Comput. Modeling 29 (1999) 229–233.[28] S. Cohen, J. Rosiński, Gaussian approximation of multivariate Lévy processes with applications to simulation of tempered stable processes, Bernoulli

13 (2007) 195–210.